Constraints on free parameters of the simplest bilepton gauge model from the neutral kaon system mass difference

Constraints on free parameters of the simplest bilepton gauge model from the neutral kaon

system mass difference

F. Pisano

Departamento de Fı´sica, Universidade Federal do Parana´, 81531-990 Curitiba, PR, Brazil

J. A. Silva-Sobrinho

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil

M. D. Tonasse*

Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro, Rua Sa˜o Francisco Xavier 524, 20550-013 Rio de Janeiro, RJ, Brazil ~Received 5 February 1998; published 29 July 1998!

We consider the contributions of the exotic quarks and gauge bosons to the mass difference between the short- and the long-lived neutral kaon states in the SU~3!C3SU~3!L3U~1!Nmodel. The lower boundMZ₈;14

TeV is obtained for the extra neutral gauge bosonZ80_{. Ranges for values of one of the exotic quark masses and}

quark mixing parameters are also presented.

@S0556-2821~98!05215-1#

PACS number~s!: 12.60.2i, 12.15.Ff, 14.70.Pw

The Dm5mK_L2mK_S mass difference between the

long-and the short-lived kaon states was successfully used in a two-generation standard model to predict the charmed quark mass @1#. In the following years several authors studied the K0-K¯0 system in order to find constraints on the parameters of new gauge theories such as gauge and scalar boson masses and mixing angles~see, for example, Refs.@2–4#!. The idea behind this is that, since thecquark mass was predicted with good precision, a possible new contribution to Dm must be smaller than the result obtained from the two generations.

In this paper we go back to this subject in order to con-strain the new neutral gauge boson mass and quark mixing parameters imposed by the SU~3!_C3SU~3!_L3U~1!_N ~3-3-1!

model@5–7#. The 3-3-1 model is a gauge theory based on the SU~3!_C^SU~3!_L^U~1!_N semisimple symmetry group. It has

the interesting feature that the anomaly cancellation does not happen within each generation, as in the standard model, but only when the three generations are considered together. Thus, the number of families must be a multiple of the color degrees of freedom and, as a consequence, the 3-3-1 model suggests a route towards the solution of the flavor question

@5–8#.

Let us summarize the most relevant points of the model. In the minimal particle content of Ref. @5# the left-handed quark fields transform under the SU~3!_Lgroup as the triplets

Q_1L5

S

uw

d1u

J1

D

_L ;~3,2

3!, QaL5

S

Jaf

uaw

dau

D

_L

;~3*,213!

~1!

(a52,3), where 2/3 and21/3 are the U~1!_N charges. Each left-handed quark field has its right-handed counterpart trans-forming as a singlet of the SU~3!_L group. In order to avoid anomalies one of the quark families must transform in a dif-ferent way with respect to the two others. In Ref. @6# the singled family is the third one, but this is not relevant here. The exotic quarkJ₁carries 5/3 units of electric charge while J2 andJ3 carry24/3 each. In the gauge sector the

single-charged (V6_{) and the double-charged (U}66_{) vector}

bilep-tons@9#, together with a new neutral gauge bosonZ

₈

0, com-plete the particle spectrum with the charged W6 _{and the}

neutral Z0 _{standard gauge bosons. At low energy the 3-3-1}

model recovers the standard phenomenology @7,10#. An important property is that the bileptons can have a low energy scale. Low energy data constrain the vector bilepton masses to MX. 230 GeV (X[V1,U11) @11#. Some

au-thors have used the running of the coupling constant to im-pose upper bounds on 3-3-1 gauge boson masses @12,13#. However, this procedure involves an arbitrary normalization of N @6,7#. Usually this is done like in the standard model, although it is not mandatory. Recently these upper bounds were reexamined and it was found that the MZ₈ mass does

not have upper bound, differently from previous calculations, while M_X,3.5 TeV@14#.

The Dm mass difference has already been studied in the context of the 3-3-1 model at the tree level~Fig. 1a!in order to put a lower bound on theM_Z8mass@15#and on the mix-ing parameters @13#, in the last case, taking into account an upper bound onMZ₈. Here we apply the experimental lower

limits and these reexamined upper bounds on the M_X gauge boson masses in order to obtain the impositions ofDm upon some of the 3-3-1 free parameters. Since the bileptons couple exotic to ordinary quarks, leading to additional contributions to the box diagram for the K0-K¯0 transition ~Fig. 1b!, we combine the tree level with the box contribution.

The charged current interactions for the quarks are given in Ref.@5#and we can rewrite them as

*Email address: Tonasse@vax.fis.uerj.br

PHYSICAL REVIEW D, VOLUME 58, 057703

L₅₂ g

2

A

2@U ¯_gm_~1

2g5!VCKMDWm11U¯gm~12g5!zJVm

1D¯gm_~1

2g5!jJUm#1H.c., ~2!

where

U5

S

u c t

D

, D5

S

d s b

D

, V_m5

S

Vm1 Um22 Um22

D

, U_m5

S

Um22 Vm1 Vm1

D

,

~3!

and J₅diag(J1 J2 J3). The VCKM is the usual

Cabibbo-Kobayashi-Maskawa mixing matrix andz andj are mixing matrices containing the new unknown parameters due to the presence of the exotic quarks. Here, unlike Eq. ~1!, we are working with the mass eigenstates.

We are using the unitary gauge. The box diagram for the K0-K¯0transition is represented in Fig. 1b, where, in the stan-dard model,qi5Ui(i51,2,3) andX is theW2boson. In the

3-3-1 model, besides these, we have new contributions: for i51, q₁5J₁ and X5U22_{, while for i5a52,3, q}

a5Ja with X5V2_{. We do not consider here contributions of the}

scalar bileptons. This would be very complicated because of the proliferation of unknown parameters as mixing angles and scalar boson masses@16#. However, because of the

elu-sivity of the scalar particles, we expect that the new contri-butions of the 3-3-1 model will be dominated by the gauge bosons.

Our calculation is standard @1#. Neglecting long-distance terms, we defineL_X—an effective Lagrangian corresponding to theXgauge boson exchange in the box diagram—as being the free quark amplitude A_X(ds¯→¯ sd ) and we evaluate the matrix element

^

K¯0_u

2L_XuK0

_&

_{. According to the MIT bag}

model we can write @4#

A_X520.7

S

GFMW

2

pM_X

D

2

O_sd

(

i,j

G_iG_j

f~a_q i

X_!

2f~a_q j

X_!

a_q i X

2a_q j

X , ~4!

where the G_i are mixing parameters given in the standard model byV_is*Vidand in the 3-3-1 model byjis*jid. From the

momentum integration we have

f~x!5₁1 2x

S

11

x2lnx

12x

D

. ~5!

In Eq. ~4!, a_q i X

5(mq_i/MX)2 is the square ratio of the mq_i

quark and the MX gauge boson masses. We have defined

O_sd[@v

s

¯_gm(12g5)u

d#2. The sum in Eq. ~4! runs over all

quarks coupled by the X gauge boson. TheDmX mass

dif-ference due to theX gauge boson contribution is given by

DmX522Re

^

K¯0uLXuK0

&

, ~6!

withL_X[AX. Taking into account the relation

^

K¯0uO_sduK0

&

54

3fK

2

mK, ~7!

wheremK5498 MeV andfK5161 MeV are the kaon meson

mass and its decay constant, respectively, the total mass dif-ference is

Dm5

8hm_K

3

S

f_KM_W

sWMX

D

2

(

i,j

G_iG_j

f~a_q i X

!2f~a_q j X

!

a_q i X

2a_q j

X , ~8!

where s_W5sinu_W,u_W is the Weinberg weak mixing angle, and we have included the leading order QCD correction fac-torh50.55@13,17#.

Despite the uncertainties undergone by the evaluation of the effective Lagrangian L_X, the application of this method to well-known processes tells us that the procedure is reliable

@3#.

In order to get physical results from Eq. ~8! we perform some hypotheses on the free parameters. First, we notice that the standard model does not give the wholeDmmass differ-ence. Considering only two quark generations we findDm₂

52310215 GeV, while the experimental value is Dmexpt 53.491310215 _{GeV @18#. The top quark contribution is}

negligible because of the smallness of the corresponding

FIG. 1. K0-K¯0transition originating theDm5m_K

L2mKS mass

difference at the tree level~a!and the box diagram~b!in the 3-3-1 model. Theq’s are all quarks coupled byX gauge bosons todand

s.

BRIEF REPORTS PHYSICAL REVIEW D58057703

mixing parameters. Thus, if the 3-3-1 model is realized in nature, we can expect that its pure contribution to Dm is about 10215 GeV.

Let us examine individual 3-3-1 contributions under the conservative hypothesis in which each one of them gives the total assumed 3-3-1 counterpart toDm, i.e., 10215GeV. We begin analyzing the contribution of theUbilepton and theJ1

quark. For convenience we assume for thej mixing matrix the same parametrization advocated by the Particle Data Group for VCKM@18#. In this case, since we choose all the

mixing angles in the first quadrant, the parameter j1*dj1s is

positive. We do not consider hereC Pviolating phases in the mixing matrices. In Fig. 2 we plot this mixing parameter as function of M_U, according to Eq.~8!, where we are using the bounds MU.230 GeV @11# and MU,3500 GeV @14#.

Hence, we can see that 0.014&j₁*_dj1s&0.35. Physically this

range for the mixing angles implies mJ₁&MU. In order to

apply the lower limit on the mixing parameter for obtaining a lower bound forMZ₈, we use the result of Ref.@15#for the

Z

₈

contribution to Dm, but taking into account that we are assuming that the maximal contribution to Dm is ;10215 GeV ~not the whole experimental value!. We introduce also the leading order QCD correction factorh50.55. This leads to MZ₈;1.033103@Re(VL

D

11

*V_LD12)#1/2 TeV, where VL D_{, the}

mixing matrix relating the symmetry left-handed quark states carrying21/3 units of electric charge (D_L

8

) with the physical ones (DL), is defined by DL

8

5VL

D

DL. Since only the J1

quark carries 5/3 units of electric charge, does it not mix. Therefore, the mixing parameter V_LD₁₁*V_LD12 is the same as

j₁*_sj1d, whose lower bound getsMZ

8

;14.4 TeV.

A more rigorous analysis, considering the contributions of diagrams exchanging the single-charged V bilepton and J2

andJ3 exotic quarks, is complicated since these two quarks

can mix and the sign of the mixing parameters is not defined according to our parametrization of the mixing matrices and the choice of the mixing angles. However, it is not expected to make an appreciable difference in the results.

In this Brief Report we have studied the implications of the Dm mass difference on free parameters of the 3-3-1 model. However, differently from previous calculations

@13,15#, here we are taking into account the contribution of the box diagram exchanging the double-chargedU22

bilep-ton. Our results differ from the previous ones because we have combined the analysis for the tree level and the box diagram. Therefore, if the 3-3-1 Higgs contribution toDm is not important, we can assume the value we have estimated forMZ₈as an approximate lower bound~i.e.,MZ₈*14 TeV!.

We stress that this bound does not depend on the aforemen-tioned arbitrary normalization of N. The crucial parameter for this value is the experimental lower bound on the mass of the U22 _{bilepton gauge boson.}

We would like to thank Dr. M. C. Tijero for reading the manuscript and the Instituto de Fı´sica Teo´rica, UNESP, for use of its facilities. The work of M. D. T. was supported by the Fundac¸a˜o de Amparo a` Pesquisa no Estado do Rio de Janeiro ~proc. No. E-26/150.338/97!.

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mass. The dark region represents the allowed values forj_1d*j_1s.

BRIEF REPORTS PHYSICAL REVIEW D58057703